or are they a private entity?
I guess what I'm getting at is the fact that IMHO you have level's of organizations....
1) The church itself & other ecclesiastical bodies (the parish, bishop's, etc.), sisterhoods, monastery's, etc. These generally require all employees to be Catholic with possibly very few exceptions (secretary, etc.). The primary purposes of these institutions is religious.
2) Catholic owned / affiliated organizations -- parochial schools; Universities such as Notre Dame, Boston U, Loyola, etc.; Catholic hospitals, etc. -- These organizations are ultimately owned or controlled by #1 above.
There are either no restrictions or relatively few restrictions on the religion of employees. For instance, our local parochial schools do not require you to be catholic to teach at them; however, but you do have a morals clause that you cannot live with a non-married partner, etc. In addition, catholic employees must be in good standing with their parish.
3) Other businesses owned by catholic businessmen, etc. This could be a burger king owned by a catholic.
Obama exempted #1 from the requirement (which the first amendment would have exempted anyway), but applied it to #2 and #3. Its settled law that #3 is cannot wiggle out of it due to "religious persecution" -- see civil rights laws, etc.
#2 could be the sticking point, and I really don't know. For instance, our local Catholic law school has a Jewish Dean, and a LGBT club. There is no way IMHO they should be exempted. On the other hand I could see parochial schools having a stronger case.